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What is F1​, the force on the ball due to the bottom spring, at t=0? Your answer must be a vector with appropriate units.

What is F1​, the force on the ball due to the bottom spring, at t=0? Your answer must be a vector with appropriate units.

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What is F1​, the force on the ball due to the bottom spring, at t=0? Your answer must be a vector with appropriate units.

### Understanding the Dynamics of a Mass-Spring System #### System Overview A ball of mass \( m = 5 \, \text{kg} \) is held motionless at \( t = 0 \) between two vertical springs. Both springs have a relaxed length \( L_0 = 3 \, \text{m} \). The springs connect to each other at the center of the ball. #### Specifications 1. **Bottom Spring (Orange)** - **Fixed Point:** Its fixed end is positioned on the floor, which is defined as the origin of the coordinate system. - **Stiffness:** \( k_1 = 300 \, \text{N/m} \) - **Current State:** It is compressed to a length \( L_1 = 2.5 \, \text{m} \). 2. **Top Spring (Green)** - **Fixed Point:** Its fixed end is attached to the ceiling. - **Stiffness:** \( k_2 = 100 \, \text{N/m} \) - **Current State:** It is stretched to a length \( L_2 = 4.5 \, \text{m} \). 3. **Ceiling Height** - The ceiling is at a height \( h = L_1 + L_2 = 7 \, \text{m} \) above the floor. - Gravity acts in the downward direction. #### Diagram Explanation The accompanying diagram depicts the system as follows: - A ball (denoted as \( m \)) is located between two springs. - The bottom spring, shown in orange, is compressed. - The top spring, shown in green, is stretched. - The coordinate axes \( x \) and \( y \) are shown in pink, with \( y \) pointing upwards and \( x \) to the right. - The fixed endpoints of the springs are black lines representing the ceiling and the floor. The diagram helps visualize the mechanical equilibrium of the system, where the forces due to the springs balance the gravitational force acting on the mass. --- This setup is commonly used in physics to understand how springs and masses interact under forces such as gravity, providing insight into concepts like Hooke’s Law, equilibrium, and energy transfer in oscillatory systems.
Transcribed Image Text:### Understanding the Dynamics of a Mass-Spring System #### System Overview A ball of mass \( m = 5 \, \text{kg} \) is held motionless at \( t = 0 \) between two vertical springs. Both springs have a relaxed length \( L_0 = 3 \, \text{m} \). The springs connect to each other at the center of the ball. #### Specifications 1. **Bottom Spring (Orange)** - **Fixed Point:** Its fixed end is positioned on the floor, which is defined as the origin of the coordinate system. - **Stiffness:** \( k_1 = 300 \, \text{N/m} \) - **Current State:** It is compressed to a length \( L_1 = 2.5 \, \text{m} \). 2. **Top Spring (Green)** - **Fixed Point:** Its fixed end is attached to the ceiling. - **Stiffness:** \( k_2 = 100 \, \text{N/m} \) - **Current State:** It is stretched to a length \( L_2 = 4.5 \, \text{m} \). 3. **Ceiling Height** - The ceiling is at a height \( h = L_1 + L_2 = 7 \, \text{m} \) above the floor. - Gravity acts in the downward direction. #### Diagram Explanation The accompanying diagram depicts the system as follows: - A ball (denoted as \( m \)) is located between two springs. - The bottom spring, shown in orange, is compressed. - The top spring, shown in green, is stretched. - The coordinate axes \( x \) and \( y \) are shown in pink, with \( y \) pointing upwards and \( x \) to the right. - The fixed endpoints of the springs are black lines representing the ceiling and the floor. The diagram helps visualize the mechanical equilibrium of the system, where the forces due to the springs balance the gravitational force acting on the mass. --- This setup is commonly used in physics to understand how springs and masses interact under forces such as gravity, providing insight into concepts like Hooke’s Law, equilibrium, and energy transfer in oscillatory systems.
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